The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Nature

One can tell by the name itself what chaos theory might mean. Chaos means something that is unpredictable, random and unstable. There are many known and predictable phenomena in science such as electricity, gravity or chemical reactions; however, chaos theory examines things that are not possible to control. For example, nature: weather, earthquakes, clouds, trees, tsunamis, and tornadoes. Other than nature, there are human-related unpredictable things, such as the stock market and our brain states. Chaos theory is a field of mathematics that deals with complex systems whose behavior is highly sensitive to the slightest changes in initial conditions. For example, someone clapping their hands could change the weather, so even the smallest alterations can have big consequences.

Chaos theory emerged around the second half of the 20th century. This is because chaos theory has complex systems and these systems contain many elements that move. For this reason, computers are needed to calculate all the different possibilities. How did chaos theory come to be? A man named Edward Lorenz, a meteorologist, created a weather model on his computer in 1960. This weather model consisted of an extensive array of complex equations to predict weather conditions. This model always gave different sequence of numbers that represented weather conditions. One day he became curious and ran his own tests to see what the outcome would be. After running a sequence, he started running the same sequence halfway through, re-entering the numbers the first sequence had given him at that point. The results were not what he was expecting; they were entirely different from his first outcomes. The second time, he entered numbers that were rounded to three instead of six digits (for example, .506 versus .506127). Since the difference between these numbers is not much, he expected the results to be only slightly varied. However, that small error gave completely different outcome. Form this he concluded that even the slightest differences in initial conditions makes prediction of past or future outcomes impossible.

Image: J.L.Westover.

There are many principles of Chaos. One of them is the butterfly effect, also described by Lorenz. It is said that even a small butterfly flapping its wings in America can create a hurricane in Japan; if the butterfly did not flap its wing at the “right” time in space then the hurricane would not have happened. Even the smallest behavior has a direct effect in the future. Another principle is unpredictability. Since it is not possible to know all the initial conditions of a complex system in adequate detail, we can’t possibly know the outcome of those. As explained above, even the smallest change in numbers can lead to a big errors in prediction; outcomes can be completely different from what is expected.

We can never know for certain when we might have a storm or tsunami until few days before it’s about to happen. Similarly to the weather, chaos is present in our daily life. For example, the bus you usually take was late and you decided to take another bus, and randomly you meet a person, and you both start talking, he makes an impression on you, you go on a date with him, fall in love, get married and grow old together. Now imagine that the person had a similar situation: he decided to take this bus rather than his usual bus and met you. What if he never got on that bus at the right time to meet you, and what if you had decided to wait for you usual bus? It is scary to think about how one small decision makes such a big difference in your life.

You have been told from the time you started school that math was important because math is everywhere. Did you ever believe that? The point of this post is to prove that statement. Math is everywhere, specifically the golden ratio.

The golden ratio is Φ = (1 + √5) /2 = 1.61803398874989484820. “This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section.”1 This number was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in “De Divina Proportione” in 1509, by Johannes Kepler around 1600, and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.”1

The golden ratio is obviously found in the world of mathematics. The golden ratio is created when one can divide a line in a unique way. Imagine being presented with a wooden plank to cut. Where should I make the cut? There is one unique place you could cut that would give you the golden ratio. This mean that the ratio of the larger piece to the smaller piece is the same ratio as the larger piece to the entire plank prior to being cut. We could then cut that smaller piece at a certain point and get a piece one and piece two such that the ratio of piece one to piece two is the same ratio as piece two to the whole smaller piece. And the process could continue. So what makes that so special? It is special because this proportion doesn’t just appear in mathematics; it appears in your body, nature, architecture, and the solar system.

Nature and Life

Think of an ant or search of an image for an ant. Their bodies seem a bit odd at first, but look closer. An ant’s body has been distributed by the golden ratio. Think of a moth or a butterfly. In order for their wings to do what they do, they have been distributed in the golden ratio. Think of a snail’s shell or the classic spiral seashell. How is it that the shell can look like a never-ending spiral? It is because that shell uses the golden ratio. Look at your neighbor. The torso to leg, the head to the torso, the sections of your fingers; all of these are examples of the golden ratios in your own body.

It was always said that beauty is in the eye of the beholder, but is that really true? What if I told you that beauty was based on the golden ratio? Would you believe me? There is sound basis in scientific study and evidence to support that what we perceive as beauty in women and men is based on how closely the proportions of facial and body dimensions come to Phi.2 “For this reason, Phi is applied in both facial plastic surgery and cosmetic dentistry as a guide to achieving the most natural and beautiful results in facial features and appearance.”2

Because of the constant presence of phi in nature, we, as a civilization, have brought the golden ratio into many of our masterpieces in art. Consider the Egyptian Pyramids. It is said that the ratio between the height, base, and hypotenuse is the golden ratio. Greeks were aware of the golden ratio when they built the Parthenon.1 It is quite obvious if you have seen the movie, The Da Vinci Code, that Leonardo Da Vinci used phi in his classic drawings. The painting of the Last Supper used the golden ratio to determine the placement of Christ and the disciples to the table, walls, and windows around them.1

The golden proportion is in places where we would never imagine. Next time you buy a bottle of Pepsi, look very critically. How did they decide where to but the circular logo or how big to make the logo or to what proportion the writing should be to the logo? These questions are answered by using the golden ratio. “It’s even being used in high fashion clothing design, such as in the “Phi Collection” announced in 2004 and covered by Vogue, Elle and Vanity Fair.”1

“In matters of reason, seeing is believing but in matters of faith, it is believing that first opens the door to seeing. The best way to know for yourself where Phi is present and where it is imagined is to explore with an open mind, learn and reach your own conclusions.”1This post was created to open your eyes to being critical of how things came to be around you. One can always ask the questions “Why?”. I challenge you to ask that question and play with the idea that the golden ratio may be the answer. Your opinion is your own. I can only present you with articles that either support or deny the importance of the golden ratio. This particular post was in support of its importance, but please feel free to read the following article that has a difference perspective. http://www.umcs.maine.edu/~markov/GoldenRatio.pdf